In some of our constructions, the main ingredients come from chiral conformal field theory.
Let us summarize here its operator-algebraic treatment (see also [1]}, [2]}).
An open, connected, non dense and non empty subset \(I\)  of \(S^1\)  is called an interval
in \(S^1\) . We identify \({\mathbb {R}}\)  as a dense subset in \(S^1\)  by the stereographic projection.
The Möbius group \({\rm PSL}(2,{\mathbb {R}})\)  acts on \(S^1 = {\mathbb {R}}\cup \lbrace \infty \rbrace \)  and under this identification it contains
translations and dilations of \({\mathbb {R}}\) .
A (Möbius covariant) net (of von Neumann algebras) on \(S^1\) is an assignment of
von Neumann algebras \({\mathcal {A}}_0(I)\)  on a common Hilbert space \({\mathcal {H}}_0\)  to intervals \(I\) 
such that
